3 3 FOREWORD Quantum Chromo-Dynamics (QCD), and more generally the physics of the Standard Model (SM), enter in many ways in high energy processes at TeV Colliders, and especially in hadron colliders (the Tevatron at Fermilab and the forthcoming LHC at CERN), First of all, at hadron colliders, QCD controls the parton luminosity, which rules the production rates of any particle or system with large invariant mass and/or large transverse momentum. Accurate predictions for any signal of possible New Physics sought at hadron colliders, as well as the corresponding backgrounds, require an improvement in the control of uncertainties on the determination of PDF and of the propagation of these uncertainties in the predictions. Furthermore, to fully exploit these new types of PDF with uncertainties, uniform tools (computer interfaces, standardization of the PDF evolution codes used by the various groups fitting PDF s) need to be proposed and developed. The dynamics of colour also affects, both in normalization and shape, various observables of the signals of any possible New Physics sought at the TeV scale, such as, e.g. the production rate, or the distributions in transverse momentum of the Higgs boson. Last, but not least, QCD governs many backgrounds to the searches for this New Physics. Large and important QCD corrections may come from extra hard parton emission (and the corresponding virtual corrections), involving multi-leg and/or multi-loop amplitudes. This requires complex higher order calculations, and new methods have to be designed to compute the required multi-legs and/or multi-loop corrections in a tractable form. In the case of semi-inclusive observables, logarithmically enhanced contributions coming from multiple soft and collinear gluon emission require sophisticated QCD resummation techniques. Resummation is a catch-all name for efforts to extend the predictive power of QCD by summing the large logarithmic corrections to all orders in perturbation theory. In practice, the resummation formalism depends on the observable at issue, through the type of logarithm to be resummed, and the resummation methods. In parallel with this perturbative QCD-oriented working programme, the implementation of both QCD/SM and New physics in Monte Carlo event generators is confronted with a number of issues which deserve uniformization or improvements. The important issues are: 1) the problem of interfacing partonic event generators to showering Monte-Carlos; 2) an implementation using this interface to calculate backgrounds which are poorly simulated by the showering Monte-Carlos alone; 3) a comparison of the HERWIG and PYTHIA parton shower models with the predictions of soft gluon resummation; 4) studies of the underlying events at hadron colliders to check how well they are modeled by the Monte-Carlo generators. In this perspective, our Working Group devoted its activity to improvements of the various QCD/SM ingredients relevant both for searches of New Physics and estimates of the backgrounds to the latter at TeV colliders. This report summarizes our work. Section 1 reports on the effort towards precision Parton Distribution Functions (PDF s). Section 2 presents the issues worked out along the two current frontiers of Higher Order QCD calculations at colliders, namely the description of multiparton final states at Nextto-Leading Order (NLO), and the extension of calculations for precison observables beyond this order. Section 3 resummarizes 1 a large variety of questions concerning the relevance of resummation for observables at TeV colliders. In parallel with these general purpose tackling angles, more specific studies, dedicated to jet physics and improved cone algorithms, and to the QCD backgrounds to H γγ, both irreducible (isolated prompt photon pairs) and reducible (photon pion), are presented in section 4. Finally, section 5 summarizes the activities of the Intergroup on Monte Carlo issues, which are of practical interest for all three Working Groups of the Workshop: HIGGS [1], BSM [2] and the present one. 1 E. Laenen c, all rights reserved.

4 4 1. PARTON DISTRIBUTION FUNCTIONS 2, 3 The experimental uncertainties in current and future hadronic colliders are decreasing to a level where more careful consideration has to be given to the uncertainties in the theoretical predictions. One important source of these uncertainties has its origin in Parton Distribution Functions (PDFs). The PDF uncertainties in turn are a reflection of those in the experimental data used as an input to the PDF fits and in the uncertainties of the theoretical calculations used to fit those data. As a consequence, sophisticated statistical methods and approximations have to be developed to handle the propagation of uncertainties. We will give a summary of the current status of several methods being pursued. To fully exploit these new types of PDF fits a uniform computer interface has been defined and developed. This code provides easy access to all the present and future fits. The code is available from the website pdf.fnal.gov. Such an interface is most efficient if the evolution codes of the various groups fiiting the PDFs are standardized to a sufficient degree. For this purpose detailed and accurate reference tables for the PDF evolution are provided at the end of this section. 1.1 Methods for estimating parton distribution function uncertainties A mathematical framework At first sight PDF fitting is rather straightforward. However, a more detailed look reveals many difficult issues. As the PDF uncertainties will affect all areas of phenomenology at hadron colliders, a clear mathematical framework of a PDF fit is essential [3]. From this formulation, all explicit methods can be derived. Also, the mathematical description will make explicit all assumptions needed before one can make a fit. These assumptions are crucial and do not find their origin in experimental results, but rather in theoretical prejudice. Such assumptions are unavoidable as we fit a system with an infinite number of degrees of freedom (the PDF functional) to a finite set of data points. We want to calculate the probability density function Ppdf O which reflects the uncertainty in predicting the observable O due to the PDF uncertainties. The function Ppdf O (x e) gives the probability density to measure a value x e for observable O. To calculate the PDF probability density function for observable O we have to integrate over the functional space of all possible PDFs V (F). The integration is weighted by three probability density functions: the prior probability density function, P prior, the experimental response function of the observable, Pexp O input and the probability density function of the fitted experiments, Pexp. The resulting formula is given by Ppdf O (x e) = d F P prior (F) Pexp input (F) Pexp O (x e x t (F)). (1) V (F) The prior probability density function contains theoretical constraints on the PDF functionals such as sum rules and other potential fit constraints (e.g. an explicit α S (M Z ) value). The most crucial property of the prior function is that it defines the functional integral by imposing smoothness constraints to make the number of degrees of freedom become finite. The simplest example is an explicit finite parametrization of the PDF functionals Ppdf O (x e) = dλ 1 dλ 2... dλ n P prior ({λ}) Pexp input ({λ}) Pexp O (x e x t ({λ})), (2) V ({λ}) where the PDF parameters are given by the list {λ}. Note that through the functional parametrization choice we have restricted the integration to a specific subset of potential PDF functionals F. Such a 2 Section coordinators: A. Vogt, W. Giele 3 Contributing authors: S. Alekhin, M. Botje, W. Giele, J. Pumplin, F. Olness, G. Salam, A. Vogt 4 Contributing authors: S. Alekhin, M. Botje, W. Giele, J. Pumplin, F. Olness

5 5 choice is founded on physics assumptions with no a priori justification. The resulting phenomenology depends on this assumption. The experimental response function forms the interface with the experimental measurement. It gives the probability density to measure a value x e given a prediction x t. The experimental response functions contain all information about the experiment needed to analyze the measurement. The prediction x t is an approximation using a perturbative estimate of the observable amended with relevant nonperturbative corrections. A simple example would be Pexp(x O e x t ) = 1 ( δ 2π exp 12 ) (x e x t ) 2 /δ 2, where we have a Gaussian response function with a one sigma uncertainty δ. Note that the form of the response function depends on the actual experiment under consideration. It is sometimes convenient to get a result that is independent of an experiment and its particular detector: To obtain the theoretical prediction for the probability density function of the observable, one can simply replace the experimental response function by a delta function (i.e. assume a perfect detector) P theory exp (x e x t ) = δ(x e x t ). Finally the probability function for the fitted experiments is simply a product of all the experimental response functions N exp Pexp input (F) = P exp (i) (x(i) m x(i) t (F)), (3) i where x (i) m denotes the set of measurements provided by experiment (i). This function measures the probability density that the theory prediction based on PDF F describes the combined experimental measurements. Often the experimental uncertainties can be approximated by a Gaussian form of the experimental reponse function, i.e. a χ 2 description of the uncertainties: Ppdf O (x e) dλ 1 dλ 2... dλ n e 1 2 i χ2 i (x(i) m x (i) t ({λ})) e 1 2 χ2 O (x(i) e x t({λ})), (4) V ({λ}) where we have chosen a specific parametrization for the PDF functionals. This approximation leads to a more traditional approach for determining PDFs with uncertainties. These methods are outlined in sections 1.14 and To go beyond the Gaussian approximation more elaborate methods are needed. Sections 1.12 and 1.13 describe techniques to accomplish this Random sampling method This method attempts to calculate Eq. (1) without any approximations by using random sampling, i.e. a Monte Carlo evaluation of the integral [3]. By generating a sufficiently large sample of PDFs Eq. (1) can be approximated by P O pdf (x e) 1 N N i=1 P prior (F i ) P input exp (F i ) P O exp (x e x t (F i )). (5) The simple implementation of Eq. (5) would lead to a highly inefficient random sampling and an unreasonably large number of PDFs would be required. By using an optimization procedure, this problem

6 6 can be solved. The optimization procedure on the functional level of Eq. (1) is simply redefining the PDF F to F u such that the Jacobian of the transformation equals so that d F u d F = P prior(f) P input exp (F), (6) Ppdf O (x e) = V (F u ) d F u P O exp (x e x t (F u )). (7) Applying this to the random sampling evaluation gives P O pdf (x e) 1 N N Pexp O (x e x t (Fi u )). (8) i=1 In the random sampling approximation the redefined PDFs Fi u are easily identified as the unweighted PDFs with respect to the combined probability density P prior (F) Pexp input (F). That is, the density of Fi u is given by this combined probability density. As such, each of the unweighted PDFs is equally likely. This is reflected in Eq. (8), as the probability density function of the observable is the average of the response function over the unweighted PDFs. Finally, we have to generate the set {Fi u }. One method is to use the Gaussian approximation for Eq. (6) simplifying the generation of the set {Fi u} [4]. Another more general approach is to apply a Metropolis Algorithm on the combined probability density function P prior (F) Pexp input (F). This approach will handle any probability function. Furthermore, in a Metropolis Algorithm approach convergence does not depend on the number of parameters used in the PDF parametrization. Also, complicated non-linear parameter subspaces which have constant probability will be modelled correctly. These properties make it possible to use large number of parameters and explore the issue of parametrization dependence of the PDFs. Once the set {Fi u } is generated we can predict the probability density function for any observable by averaging over the experimental response function using Eq. (8) Lagrange multiplier method The values of the fit parameters that minimize χ 2 provide by definition the best fit to the global set of data. The dependence of χ 2 on those parameters in the neighborhood of the minimum can be characterized in quadratic approximation by the matrix of second derivatives, which is known as the Hessian matrix. The inverse of the Hessian matrix is the error matrix, and it forms the basis for traditional estimates of the uncertainties. The traditional assumption of quadratic behavior of χ 2 in all directions in parameter space is not necessarily a very good one in the case of PDF fitting, because there are flat directions in which χ 2 changes very slowly, so large changes in certain combinations of parameters are not ruled out. This difficulty is always present in the case of PDF fitting, because as more data become available to pin down the PDFs better, more flexible parametrizations of them are introduced to allow ever-finer details to be determined. To some extent, the flat directions can be allowed for by an iterative method [5, 6], whereby the eigenvector directions of the error matrix and their eigenvalues are found using a convergent series of successive approximations. This iterative method is implemented as an extension to the standard minimization program minuit. The result is a collection of PDFs (currently 40 in number) that probe the allowed range of possibilities along each of the eigenvector directions. The PDF uncertainty on any physical quantity or on some feature of the PDFs themselves can easily be computed after that quantity is evaluated for each of the eigenvector sets. This method has been applied, for example, to find the uncertainty range for predictions of W and Z cross sections and their correlations [6].

7 7 To completely overcome the need to assume quadratic behavior for χ 2 as a function of the fitting parameters, one can use a Lagrange Multiplier method [5, 7] to directly examine how χ 2 varies as a function of any particular variable that is of interest. The method is a classical mathematical technique that is more fully called Lagrange s method of undetermined multipliers. To explain it by example, suppose we want to find the effect of PDF uncertainties on the prediction for the Higgs boson production cross section. Instead of finding the PDF parameters that minimize χ 2 (which measures the quality of fit to the global data), one minimizes χ 2 + λσ H, where σ H is the predicted Higgs cross section which is of course also a function of the PDF parameters. The minimization is carried out for a variety of values of the Lagrange Multiplier constant λ. Each minimization provides one point on the curve of χ 2 versus predicted σ H. Once that curve has been mapped out, the uncertainty range is defined by the region for which the increase in χ 2 above its minimum value is acceptable. The essential feature of the Lagrange Multiplier method is that it finds the largest possible range for the predictions of a given physical quantity, such as σ H, that are consistent with any given assumed maximum increase χ 2 above the best-fit value of χ 2. This method has been applied, for example, to study the possible range of variation of the rapidity distribution for W production, by extremizing various moments of that distribution [5, 7] Covariance matrix method The covariance matrix method is based on the Bayesian approach to the treatment of the systematic errors, when the latter are considered as a random fluctuation like the statistical errors. Having no place for the detailed discussion of the advantages of this approach we refer to the introduction into this scope given in Ref. [8]; the only point we would like to underline here is that application of the Bayesian approach is especially justified in the analysis of the data sets with the numerous sources of independent systematic errors, which is the case for the extraction of PDFs from existing experimental data. Let the data sample have one common additive systematic error 5. Bayesian approach the measured values y i are given by In this case following the y i = f i + µ i σ i + λs i, (9) where f i (θ 0 ) is the theoretical model describing the data, θ 0 the fitted parameter of this model, σ i statistical errors, s i systematic errors for each point, µ i and λ the independent random variables, and the index i runs through the data points from 1 to N. The only assumption we make is that the average and the dispersion of these variables are zero and unity respectively. It is natural to assume that the µ i are Gaussian distributed when the data points are obtained from large statistical samples. Such an assumption is often not justified for the distribution of λ Within the covariance matrix approach the estimate of the fitted parameter ˆθ is obtained from a minimization of the functional χ 2 (θ) = N (f i (θ) y i )E ij (f j (θ) y j ), (10) i,j=1 where E ij = 1 ( δ ij ρ ) iρ j σ i σ j 1 + ρ 2, is the inverse of the covariance matrix C ij = s i s j + δ ij σ i σ j, (11) 5 We consider the case of one source of systematic errors, generalization on the many sources case is straightforward.

8 8 ρ is modulus of the vector ρ with N components equal to ρ i = s i /σ i and δ ij is the Kronecker symbol. We underline that with the data model given by Eq. (9) one does not need to assume a specific form for the distribution of y i in order to obtain the central value of this estimate. In a linear approximation for f i (θ) one also does not need such assumptions to estimate the dispersion. In this approximation the dispersion reads [9] D C (ˆθ) = 1 ρ [1 2 z 2 ] φ ρ 2 (1 z 2, (12) ) where φ is modulus of the vector φ with N components equal to φ i = f i (θ 0)/σ i, the symbol denotes the derivative with respect to θ, and z is the cosine of the angle between ρ and φ. To obtain the distribution of ˆθ one needs to know the complete set of its moments, which, in turn, requires similar information for the moments of y i. At the same time from considerations similar to the proof of the Central Limit theorem of statistics (see. Ref. [10]) one can conclude that the distribution of ˆθ is Gaussian, if all independent systematic errors are comparable in value and their number is large enough. More educated guesses of the form of the distribution can be performed with the help of the general approach described in subsection The dispersion of fitted parameters obtained by the covariance method is different from the one obtained by the offset method described in subsection Indeed, the dispersion of the parameter estimate obtained by the offset method applied to the analysis of the data set given by Eq. (9) is equal [9] D O (ˆθ) = 1 φ 2 ( 1 + ρ 2 z 2). (13) One can see that D O is generally larger than D C. The difference is especially visible for the case N 1, when ρ 1, if the systematic errors are not negligible as compared to the statistical ones. In this case and if z 1 while D C (ˆθ) 1 φ 2 (1 z 2 ), (14) D O (ˆθ) ρ2 z 2 φ 2. (15) One can see that the standard deviation of the offset method estimator rises linearly with the increase of the systematics, while the covariance matrix dispersion saturates and the difference between them may be very large. Some peculiarities arise in the case when the systematic errors are multiplicative, i.e. when the s i in Eq. (11) are given by η i y i, where η i are constants. As it was noted in Ref. [11] in this case the covariance matrix estimator may be biased. The manifestation of this bias is that the fitted curve lays lower the data points on average, which is reflected by a distortion of the fitted parameters. In order to minimize such bias one has to calculate the covariance matrix of Eq. (11) using the relation s i = η i f i (θ). In this approach the covariance matrix depends on the fitted parameters and hence has to be iteratively re-calculated during the fit. This certainly makes calculation more time-consuming and difficult, but in this case the bias of the estimator is non-negligible as compared to the value of its standard deviation if the systematic error on the fitted parameter is an order of magnitude larger than the statistical error [9] Offset Method With the offset method [12], already mentioned above, the systematic errors are incorporated in the model prediction t i (θ,λ) = f i (θ) + k λ k s ik, (16)

9 9 where we allow for several sources of systematic error (k). The χ 2, to be minimized in a fit, is defined as χ 2 (θ,λ) = ( ) yi t i (θ,λ) 2 + λ 2 k σ. (17) i i It can be shown [7] that leaving both θ and λ free in the fit is mathematically equivalent to the covariance method described in the previous section. However, there is also the choice to fix the systematic parameters to their central values λ k = 0 which results in minimizing k ( ) yi f i (θ) 2, (18) χ 2 (θ) = i σ i where only statistical errors are taken into account to get the best value ˆθ of the parameters. Because systematic errors are ignored in the χ 2 such a fit forces the theory prediction to be as close as possible to the data. The systematic errors on θ are estimated from fits where each systematic parameter λ k is offset by its assumed error (±1) after which the resulting deviations θ are added in quadrature. To first order this lengthy procedure can be replaced by a calculation of two Hessian matrices M and C, defined by M ij = 1 2 χ 2 C ij = 1 2 χ 2. (19) 2 θ i θ j 2 θ i λ j The statistical covariance matrix of the fitted parameters is then given by while a systematic covariance matrix can be defined by [13] where C T is the transpose of C. V θ stat = M 1, (20) V θ syst = M 1 CC T M 1, (21) Having obtained the best values and the covariance matrix of the parameters, the covariance of any two functions F(θ) and G(θ) can be calculated with the standard formula for linear error propagation V FG = ij F(θ) θ i V θ ij G(θ) θ j, (22) where V θ is the statistical, systematic or, if the total error is to be calculated, the sum of both covariance matrices. Comparing Eqs. (17) and (18) it is clear that the parameter values obtained by the covariance and offset methods will, in general, be different. This difference is accounted for by the difference in the error estimates, those of the offset method being larger in most cases. In statistical language this means that the parameter estimation of the offset method is not efficient. The method has a further disadvantage that the goodness of fit cannot be directly judged from the χ 2 which is calculated from statistical errors only. For a global QCD analysis of deep inelastic scattering data which uses the offset method to propagate the systematic errors, we refer to [14] (see for the corresponding PDF set with full error information).

10 The LHAPDF interface Introduction The Les Houches Accord PDF (LHAPDF) interface package is designed to work with PDF sets. A PDF set can consist of many individual member PDFs. While the interpretation of the member PDFs depends on the particular set, the LHAPDF interface is designed to accommodate PDFs with uncertainties as well as central fit PDFs. For PDFs with uncertainties the PDF set represents one fit to the data. For instance, a random sampling PDF set would use Eq. (8). In other words for each PDF in the set the observable is calculated. The set of resulting predictions of the observable build up the probability density. The individual member PDFs of the set are needed to calculate the PDF uncertainty on the observable. All PDF sets are defined through external files. This means that a new set can be added by simply downloading its file while the LHAPDF interface code does not change. The evolution code is not part of LHAPDF. The current default choice included and interfaced is QCDNUM [15]. Each group that contributes PDF sets can provide their own evolution code; or they can employ QCDNUM, which is available in the package The philosophy The Les Houches Accord Parton Distribution Function interface was conceived at the Les Houches 2001 workshop in the PDF working group to enable the usage of Parton Distribution Functions with uncertainties in a uniform manner. When PDFs with uncertainties are considered, a fit to the data no longer is described by a single PDF. Instead in its most flexible implementation, a fit is represented by a PDF set consisting of many individual PDF members. Calculating the observable for all the PDF members enables one to reconstruct the uncertainty on the observable. The LHAPDF interface was made with this in mind and manipulates PDF sets. The LHAPDF interface can be viewed as a successor to PDFLIB and improvements were added. To list some of the features: The handling of PDF sets to enable PDF fits that include uncertainties. The default evolution codes are benchmarked and compared for accuracy. Apart from accuracy another important feature of the evolution code is speed. Currently the default for evolution program is QCDNUM. All PDF sets are defined through external files in parametrized form. This means the files are compact compared to storing the PDFs in a grid format. Also new PDF sets can be defined by constructing the PDF defining files. The actual LHAPDF code does not have to be changed. The LHAPDF code is modular and default choices like the QCDNUM evolution code can be easily altered. Note that the current best fit PDFs can be viewed as PDF sets with one member PDF and can be easily defined through the PDF set external file definition. Alternatively one can group these fits is single sets (e.g. MRST98 set) as they often represent best fits given different model assumptions and as such reflect theoretical modelling uncertainties. The first version of the code is available in Fortran from Interfacing with LHAPDF The interface of LHAPDF with an external code is easy. We will describe the basic steps sufficient for most applications. The web site contains more detailed information about additional function calls. The function calls described here will be not be altered in any way in future versions. Including the LHAPDF evolution code into a program involves three steps: 1. First one has to setup the LHAPDF interface code: 6 Contributing authors: S. Alekhin, W. Giele, J. Pumplin

11 P t b c s ū d g d u s c b t P t b c s u d g d ū s c b t Table 1: The flavor enumeration convention used in the LHAPDF interface. (Note: CTEQ code use a different labelling scheme internally, with 1 2 and -1-2, but will adopt the above standard in the Les Houches interface.) call InitPDFset(name) It is called only once at the beginning of the code. The string variable name is the file name of the external PDF file that defines the PDF set. For the standard evolution code QCDNUM it will either calculate or read from file the LO/NLO splitting function weights. The calculation of the weights might take some time depending on the chosen grid size. However, after the first run a grid file is created. Subsequent runs will use this file to read in the weights so that the lengthy calculation of these weights is avoided. The file depends on the grid parameters and flavor thresholds. This means different PDF sets can have different grid files. The name of the grid file is specified in the PDF setup file. 2. To use a individual PDF member it has to be initialized: call InitPDF(mem) The integer mem specifies the member PDF number. This routine only needs to be called when changing to a new PDF member. The time needed to setup depends on the evolution code used For QCDNUM the grid size is the determining factor. Note that mem=0 selects the best fit PDF. 3. Once the PDF member is initialized one can call the evolution codes which will use the selected member. The function call function alphaspdf(q) returns the value of α S (Q) at double precision scale Q. Note that its value can change between different PDF members. The subroutine call call evolvepdf(x,q,f) returns the PDF momentum densities f (i.e. x PDF number density) at double precision momentum fraction x and double precision scale Q. The double precision array f(-6:6) will contain the momentum PDFs using the labelling convention of table 3. As long as the member PDF is not changed (by the call InitPDF of step 2) the evolution calls will always use the same PDF member. A few additional calls can be useful: To get the number of PDF members in the set: call numberpdf(nmem). The integer Nmem will contain the number of PDF members (excluding the special best fit member, i.e. the member numbers run from 0 to Nmem). Optionally the different PDF members can have weights which are obtained by: call weightpdf(wgt). The double precision variable wgt is for unweighted PDFs set to 1/Nmem such that the sum of all PDF member weights is unity. For weighted sets the use of the weights has to be defined by the method description. To get the evolution order of the PDFs: call GetOrderPDF(order). The integer variable order is 0 for Leading Order, 1 for Next-to-Leading Order, etc. To get the evolution order of α S : call GetOrderAs(order).

12 12 The integer variable order is 0 for Leading Order, 1 for Next-to-Leading Order, etc. It is possible that during the PDF fitting the renormalization scale was chosen different from the factorization scale. The ratio of the renormalization scale over the factorization scale used in the fit can be obtained by call GetRenFac(muf). The double precision variable muf contains the ratio. Usually muf is equal to unity. To get a description of the PDF set: call GetDesc(). This call will print the PDF description to the standard output stream The quark masses can be obtained by: call GetQmass(nf,mass). The mass mass is returned for quark flavor nf. The quark masses are used in the α S evolution. The flavor thresholds in the PDF evolution can be obtained by: call GetThreshold(nf,Q). The flavor threshold Q is returned for flavor nf. If Q=-1d0 flavor is not in the evolution (e.g. the top quark is usually not included in the evolution). If Q=0d0 flavor is parametrized at the parametrization scale. For positive non-zero values of Q the value is set to the flavor threshold at which the PDF starts to evolve. The call returns the number of flavors used in the PDF: call GetNf(nfmax). Usually the returned value for nfmax is equal to five as the top quark is usually not considered in the PDFs An example A very simple example is given below. It accesses all member PDFs in the set mypdf.lhpdf and print out the α S (M Z ) value and the gluon PDF at several (x,q) points. * * program example implicit real*8(a-h,o-z) character*32 name real*8 f(-6:6) name= mypdf.lhpdf call InitPDFset(name) QMZ=91.71d0 write(*,*) call numberpdf(n) do i=1,n write(*,*) call InitPDF(i) write(*,*) PDF set,i write(*,*) a=alphaspdf(qmz) write(*,*) alphas(mz) =,a write(*,*) write(*,*) x*gluon write(*,*) x Q=10 GeV Q=100 GeV Q=1000 GeV do x=0.01d0,0.095d0,0.01d0

14 14 The generalization to higher orders is straightforward but not required for the present calculations. The LO and NLO coefficients β 0 and β 1 of the β-function in Eq. (24) and the corresponding splitting functions P (0) (x) and P (1) (x) in Eq. (25) have been known for a long time, see Ref. [17] and references therein. In the MS scheme adopted here, the coefficient β 2 has been calculated in Refs. [18, 19]. The NNLO quantities P (2) (x) have not been computed so far, but approximate expressions have been constructed [20 22] from the available partial results [23 31]. An obvious and widespread approach to Eq. (23) is the direct numerical solution by discretization in both x and µ 2 f. This method is also used by one of us (G.S.). The parton distributions are represented on a grid with n points uniformly spaced in ln 1/x. Their values at arbitrary x are defined to be equal to a p th order interpolation of neighbouring grid points. The splitting functions can then be represented as sparse matrices acting on the vector of grid points. At initialization the program takes a set of subroutines for the splitting functions and calculates the corresponding matrices with a Gaussian adaptive integrator. At each value of µ 2 f the derivatives of the parton distributions are calculated through matrix multiplication and the evolution is carried out with a Runge-Kutta method. The algorithm has partial overlap with those of Refs. [15, 32 34] and is described in more detail in Appendix F of Ref. [35]. For the reference tables presented below, the program has been run with 7 th order interpolation in x and multiple x-grids: one for 10 8 < x < 1 with 750 points, another for < x < 1 with 240 points and a third for 0.6 < x < 1 with 180 points. A grid uniform in ln µ 2 f has been used with 220 points in the range 2 GeV 2 < µ 2 f < 106 GeV 2. Halving the density of points in both x and µ 2 f leaves the results unchanged at the level of better than 1 part in 10 5 in the range 10 8 < x < 0.9 (except close to sign-changes of parton distributions). An important alternative to this direct numerical treatment of Eq. (23) is the Mellin-N moment solution in terms of a power expansion. This method is employed by the second author (A.V.). Here Eq. (23) is transformed to N-space (reducing the convolution to a simple product) and µ 2 f is replaced by a s as the independent variable, assuming that µ f /µ r is a fixed number. Expanding the resulting r.h.s. into a power series in a s, one arrives at with df p (N,a s ) da s R (0) pp (N) = 1 β 0 P (0) pp (N), = l=0 R (k 1) pp a l 1 s R (l) pp (N)f p (N,a s ) (26) = 1 β 0 P (k) pp (N) k l=1 β l β 0 R (k l) pp (N). (27) At N m LO only the coefficients β l m and P (l m) are retained in Eq. (27). The solution of Eq. (26) can be expressed as an expansion around the LO result ] ( ) R0 (N) f(n,µ 2 f [1 [ ) = + a k s U as ] 1 k(n) 1 + a k 0 U k(n) f(n,µ 2 f,0 ), (28) k=1 a 0 where µ 2 f,0 is the initial scale for the evolution, and a 0 a s(µ 2 r(µ 2 f,0 )). It is understood in Eq. (28) that the matrix structure is simplified by switching to the appropriate flavour singlet and non-singlet combinations. For the explicit construction of the remaining 2 2 matrices U k the reader is referred to Section 5 of Ref. [36]. Finally the solutions f p (N,µ 2 f ) are transformed back to x-space by f p (x,µ 2 f ) = 1 ] dz Im [e iφ x c z exp(iφ) f p (N =c+z exp(iφ), µ 2 f π ). (29) 0 The Mellin inversions (29) can be performed with a sufficient accuracy using a fixed chain of Gauss quadratures. Hence the quantities R 0 (N) and U k (N) in Eq. (28) have to the computed only once k=1

15 15 for the corresponding support points at the initialization of the program, rendering the N-space evolution competitive in speed with fast x-space codes. Except where the parton distributions become very small, an accuracy of 1 part in 10 5 or better is achieved by including contributions up to k = 15 in Eq. (28) and using at most 18 eight-point Gauss quadratures with, e.g., z max = 70, c = 2, φ = 3π/4 in Eq. (29). The two methods for solving Eq. (23) discussed above are completely equivalent, i.e., they do not differ even by terms beyond N m LO, provided that the coupling a s evolves exactly according to Eq. (24). This condition is fulfilled in the present calculations. Thus the results of our two programs can be compared directly, yielding a powerful check of the correctness and accuracy of the codes. Note that the only previously published high-precision comparison of NLO evolution programs [16] used the truncated expansion of a 2 s (µ r) in terms of inverse powers of ln µ 2 r /Λ2 which does not exactly conform to Eq. (24). Consequently such direct comparisons were not possible in Ref. [16]. Following Ref. [17], the N-space solution (28) has usually been subjected to a further expansion in the coupling constants, retaining only the terms up to order m in the product of the U-matrices. Only the terms thus kept in Eq. (28) are free from contributions by the higher-order coefficients β k>m and P (k>m), and only these terms combine to factorization-scheme independent quantities when the parton distributions are convoluted with the N m LO partonic cross sections. Reference results for the truncated solution will be presented elsewhere [37] Initial conditions and heavy-quark treatment The following initial conditions for the reference results have been set up at the Les Houches meeting: The evolution is started at µ 2 f,0 = 2 GeV2. (30) Roughly along the lines of the CTEQ5M parametrization [38], the input distributions are chosen as xu v (x,µ 2 f,0 ) = x0.8 (1 x) 3 xd v (x,µ 2 f,0 ) = x0.8 (1 x) 4 xg (x,µ 2 f,0 ) = x 0.1 (1 x) 5 (31) x d(x,µ 2 f,0 ) = x 0.1 (1 x) 6 xū(x,µ 2 f,0 ) = (1 x) x d(x,µ 2 f,0 ) xs (x,µ 2 f,0 ) = x s (x,µ2 f,0 ) = 0.2x(ū + d)(x,µ 2 f,0 ) where, as usual, q i,v q i q i. The running couplings are specified via α s (µ 2 r =2 GeV 2 ) = (32) For simplicity these initial conditions are employed regardless of the order of the evolution and the ratio of the renormalization and factorization scales. At LO this ratio is fixed to unity, beyond LO we use µ 2 r = k r µ 2 f, k r = 0.5, 1, 2. (33) For the evolution with a fixed number N f > 3 of quark flavours the quark distributions not specified in Eq. (31) are assumed to vanish at µ 2 f,0, and Eq. (32) is understood to refer to the chosen value of N f. For the evolution with a variable N f = , Eqs. (31) and (32) always refer to three flavours. N f is then increased by one unit at the heavy-quark pole masses taken as m c = µ f,0, m b = 4.5 GeV 2, m t = 175 GeV 2, (34)

16 16 i.e., the evolution is performed as discussed in Section 1.31 between these thresholds, and the respective matching conditions are invoked at µ 2 f = m 2 h, h = c, b, t. For the parton distributions these conditions have been derived in Ref. [39]. Up to N m=2 LO they read l (N f+1) i (x,m 2 h ) = l (N f) i (x,m 2 h ) + δ m2 a 2 s ANS,(2) qq,h (x) l (N f) i (x,m 2 h ) (35) where l = q, q and i = 1,... N f, and g (Nf+1) (x,m 2 h ) = g (Nf) (x,m 2 h [ ) + ] δ m2 a 2 s A S,(2) gq,h (x) Σ (Nf) (x,m 2 h ) + AS,(2) gg,h (x) g (Nf) (x,m 2 h ) (h + h) (Nf+1) (x,m 2 h ) = δ m2 a 2 [ÃS,(2) ] s hq (x) Σ (Nf) (x,m 2 h ) + ÃS,(2) hg (x) g (Nf) (x,m 2 h ) (36) with Σ (Nf) N f i=1 (q i + q i ) and h = h. The coefficients A (2) can be found in Appendix B of ref. [39] due to our choice of µ 2 f = m 2 h for the thresholds only the scale-independent parts of the expressions are needed here from where the notation for these coefficients has been taken over. The corresponding N m LO relation for the coupling constant [40, 41] is given by a (N f+1) m s (k r m 2 h ) = a (N f) s (k r m 2 h ) + n=1 ( a (N f) s (k r m 2 h ) ) n+1 n c n,l ln k r. (37) The pole-mass coefficients c n,l in Eq. (37) can be inferred from Eq. (9) of Ref. [41], where 4a (N f 1) s expressed in terms of 4a (N f) s. Note that we use a (N f+1) s (k r m 2 h ) on the r.h.s. of Eq. (36) The benchmark results We have compared the results of our two evolution programs, under the conditions specified in Section 1.32, at 500 x-µ 2 f points covering the range 10 8 x 0.9 and 2 GeV 2 µ 2 f 10 6 GeV 2. A representative subset of our results at µ 2 f = 10 4 GeV 4, a scale relevant to high-e T jets at TEVATRON and close to m 2 W, m2 Z and, possibly, m2 Higgs, is presented in Tables 2 6. These results are given in terms of the valence distributions, defined below Eq. (31), L ± d ± ū, and the quark-antiquark sums q + q q for q = s, c and, for the variable-n f case, b. For compactness an abbreviated notation is employed throughout the tables, i.e., all numbers a 10 b are written as a b. In the vast majority of the x-µ 2 f points our results are found to agree to all five figures displayed, except for the tiny NLO and NNLO sea-quark distributions at x = 0.9, in the tables. In fact, the differences for x < 0.9 are not larger than ±1 in the sixth digit, i.e, the offsets are smaller than 1 part in Entries where these residual offsets lead to a different fifth digit after rounding are indicated by the subscript. The number with the smaller modulus is then given in the tables, e.g., should be read as with an uncertainty of ±1 in the last figure. As mentioned in Section 1.31, the three-loop (NNLO) splitting functions P (2) (x) in Eq. (25) are not yet exactly known. For the NNLO reference results in Tables 5 and 6 we have resorted to the average of the two extreme approximations constructed in Ref. [22]. The remaining uncertainties of these results can be estimated by instead employing the extreme approximations itselves. Their relative effects are illustrated, also at µ 2 f = 10 4 GeV 4, in Fig. 1. The uncertainties of the NNLO evolution to this scale turn out to be virtually negligible at x 0.05 and to amount to less than ±1% down to x l=0 is

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